Algebraic Multigrid Based on Element Interpolation (AMGe)

Brezina, Marian; Cleary, A.; Falgout, Robert D.; Henson, Van Emden; Jones, Jim E.; Manteuffel, Thomas A.; McCormick, Steve; Ruge, John W.

doi:10.1137/s1064827598344303

Cited by 191 publications

(190 citation statements)

References 17 publications

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“…In the 'classical' Ruge-Stüben AMG [39] a subset of the nodes of a certain level are identified as coarse-level nodes (so-called C-nodes) and finer-level nodes (F-nodes). In the AMGe variant information of the finite element mesh is taken into account in the coarsening procedure [12]. In smoothed aggregation (SA) multigrid [51,52] nodes of a certain level are grouped into contiguous subsets, called aggregates, that form the nodes of the next coarser level.…”

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confidence: 99%

A scalable multi‐level preconditioner for matrix‐free µ‐finite element analysis of human bone structures

Arbenz

Lenthe

Mennel

et al. 2007

Numerical Meth Engineering

132

106

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Abstract. The recent advances in microarchitectural bone imaging are disclosing the possibility to assess both the apparent density and the trabecular microstructure of intact bones in a single measurement. Coupling these imaging possibilities with microstructural finite element (µFE) analysis offers a powerful tool to improve bone stiffness and strength assessment for individual fracture risk prediction.Many elements are needed to accurately represent the intricate microarchitectural structure of bone; hence, the resulting µFE models possess a very large number of degrees of freedom. In order to be solved quickly and reliably on state-of-the-art parallel computers, the µFE analyses require advanced solution techniques. In this paper, we investigate the solution of the resulting systems of linear equations by the conjugate gradient algorithm, preconditioned by aggregation-based multigrid methods. We introduce a variant of the preconditioner that does not need assembling the system matrix but uses element-by-element techniques to build the multilevel hierarchy. The preconditioner exploits the voxel approach that is common in bone structure analysis, it has modest memory requirements, while being at the same time robust and scalable.Using the proposed methods, we have solved in less than 10 minutes a model of trabecular bone composed of 247'734'272 elements, leading to a matrix with 1'178'736'360 rows, using only 1024 CRAY XT3 processors. The ability to solve, for the first time, large biomedical problems with over 1 billion degrees of freedom on a routine basis will help us improve our understanding of the influence of densitometric, morphological and loading factors in the etiology of osteoporotic fractures such as commonly experienced at the hip, the spine and the wrist.

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confidence: 99%

A scalable multi‐level preconditioner for matrix‐free µ‐finite element analysis of human bone structures

Arbenz

Lenthe

Mennel

et al. 2007

Numerical Meth Engineering

132

106

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“…Instead, it assumes that smooth errors are characterized by the spectrum of the operator. Thus, a different guiding heuristic is suggested (see [7]) requiring the interpolation to approximate well eigenvectors corresponding to small eigenvalues of the matrix. In [7] two local measures are presented which require access to the local stiffness matrices (hence the "element-based" part in the name of the method).…”

Section: Element-based Amgmentioning

confidence: 99%

“…Thus, a different guiding heuristic is suggested (see [7]) requiring the interpolation to approximate well eigenvectors corresponding to small eigenvalues of the matrix. In [7] two local measures are presented which require access to the local stiffness matrices (hence the "element-based" part in the name of the method). The purpose of these measures is to formulate a procedure of building the interpolation aiming to fulfill the heuristic principle above.…”

Section: Element-based Amgmentioning

confidence: 99%

Adaptive Algebraic Multigrid for Finite Element Elliptic Equations with Random Coefficients

Kalchev¹

2012

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This thesis presents a two-grid algorithm based on Smoothed Aggregation Spectral Element Agglomeration Algebraic Multigrid (SA-ρAMGe) combined with adaptation. The aim is to build an efficient solver for the linear systems arising from discretization of second-order elliptic partial differential equations (PDEs) with stochastic coefficients. Examples include PDEs that model subsurface flow with random permeability field. During a Markov Chain Monte Carlo (MCMC) simulation process, that draws PDE coefficient samples from a certain distribution, the PDE coefficients change, hence the resulting linear systems to be solved change. At every such step the system (discretized PDE) needs to be solved and the computed solution used to evaluate some functional(s) of interest that then determine if the coefficient sample is acceptable or not. The MCMC process is hence computationally intensive and requires the solvers used to be efficient and fast. This fact that at every step of MCMC the resulting linear system changes, makes an already existing solver built for the old problem perhaps not as efficient for the problem corresponding to the new sampled coefficient. This motivates the main goal of our study, namely, to adapt an already existing solver to handle the problem (with changed coefficient) with the objective to achieve this goal to be faster and more efficient than building a completely new solver from scratch. Our approach utilizes the local element matrices (for the problem with changed coefficients) to build local problems associated with constructed by the method agglomerated elements (a set of subdomains that cover the given computational domain). We solve a generalized eigenproblem for each set in a subspace spanned by the previous local coarse space (used for the old solver) and a vector, component of the error, that the old solver cannot handle. A portion of the spectrum of these local eigenproblems (corresponding to eigenvalues close to zero) form the coarse basis used to define the new two-level method of our interest. We illustrate the performance of this adaptive two-level procedure with a large set of numerical experiments that demonstrate its efficiency over building the solvers from scratch.

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“…Another AMG approach that has been applied with success to elasticity is AMGe [5]. This method requires access to the element stiffness matrices and attempts to improve interpolation by determining the smooth error that needs to be interpolated well (approximations of the local near-nullspace components are determined from the element stiffness matrices).…”

Section: Relevant Prior Workmentioning

confidence: 99%

Improving algebraic multigrid interpolation operators for linear elasticity problems

Baker

Kolev

Yang

2009

Numerical Linear Algebra App

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SUMMARYLinear systems arising from discretizations of systems of partial differential equations can be challenging for algebraic multigrid (AMG), as the design of AMG relies on assumptions based on the near-nullspace properties of scalar diffusion problems. For elasticity applications, the near-nullspace of the operator includes the so-called rigid body modes (RBMs), which are not adequately represented by the classical AMG interpolation operators. In this paper we investigate several approaches for improving AMG convergence on linear elasticity problems by explicitly incorporating the near-nullspace modes in the range of the interpolation. In particular, we propose two new methods for extending any initial AMG interpolation operator to exactly fit the RBMs based on the introduction of additional coarse degrees of freedom at each node. Though the methodology is general and can be used to fit any set of near-nullspace vectors, we focus on the RBMs of linear elasticity in this paper. The new methods can be incorporated easily into existing AMG codes, do not require matrix inversions, and do not assume an aggregation approach or a finite element framework. We demonstrate the effectiveness of the new interpolation operators on several 2D and 3D elasticity problems.

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Algebraic Multigrid Based on Element Interpolation (AMGe)

Cited by 191 publications

References 17 publications

A scalable multi‐level preconditioner for matrix‐free µ‐finite element analysis of human bone structures

A scalable multi‐level preconditioner for matrix‐free µ‐finite element analysis of human bone structures

Adaptive Algebraic Multigrid for Finite Element Elliptic Equations with Random Coefficients

Improving algebraic multigrid interpolation operators for linear elasticity problems

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